The Diophantine problem in Chevalley groups

Abstract

In this paper we study the Diophantine problem in Chevalley groups $G_\pi (\Phi,R)$, where $\Phi$ is an indecomposable root system of rank $> 1$, $R$ is an arbitrary commutative ring with $1$. We establish a variant of double centralizer theorem for elementary unipotents $x_\alpha(1)$. This theorem is valid for arbitrary commutative rings with $1$. The result is principle to show that any one-parametric subgroup $X_\alpha$, $\alpha \in \Phi$, is Diophantine in $G$. Then we prove that the Diophantine problem in $G_\pi (\Phi,R)$ is polynomial time equivalent (more precisely, Karp equivalent) to the Diophantine problem in $R$. This fact gives rise to a number of model-theoretic corollaries for specific types of rings.